The critical number of finite abelian groups

Abstract

Let G be an additive, finite abelian group. The critical number $\mathsf{cr}(G)$ of $G$ is the smallest positive integer $\ell$ such that for every subset $S\subset G\setminus \{0\}$ with $|S|\ge \ell$ the following holds: Every element of $G$ can be written as a nonempty sum of distinct elements from $S$. The critical number was first studied by P. Erdős and H. Heilbronn in 1964, and due to the contributions of many authors the value of $\mathsf{cr}(G)$ is known for all finite abelian groups $G$ except for $G\cong \mathbb{Z}/pq\mathbb{Z}$ where $p,q$ are primes such that $p+\lfloor 2\sqrt{p-2}\rfloor +1<q<2p$. We determine that $\mathsf{cr}(G)=p+q-2$ for such groups.